1. Technical Field
The invention relates to the processing of digital information. More particularly, the invention relates to a method and apparatus for allowing acceptable-quality reconstruction of a signal, image, spectrum, or other digital object of interest.
2. Discussion of the Prior Art
In many fields of science and technology, it is desired to obtain information about a subject by measuring a digital signal which represents that subject. Examples include: Medical imaging (see F. Natterer, The Mathematics of Computerized Tomography, J. Wiley (1986); C. L. Epstein Introduction to the mathematics of medical imaging, Upper Saddle River, N.J.: Pearson Education/Prentice Hall (2003)), Mass Spectrometry (see A. Brock, N. Rodriguez, R. N. Zare, Characterization of a Hadamard transform time-of-flight mass spectrometer, Review of Scientific Instruments 71, 1306 (2000); P. Jansson, Deconvolution: with applications in spectroscopy, New York: Academic Press (1984)), NMR Spectroscopy (see J. Hoch, A. Stern, NMR Data Processing Wiley-Liss, New York (1998)), and Acoustic Tomography (see J. Claerbout, Imaging the Earth's Interior, Blackwell Scientific Publications, Inc. (1985)). In these different cases, the measurements are intended to allow the construction of images, spectra, and volumetric images depicting the state of the subject, which may be a patient's body, a chemical in dilution, or a slice of the earth. Many other examples can be given in other fields.
Time and effort are involved in capturing data about the underlying object. For example, it can take time to perform an MRI scan of a patient, it can take time to perform a 3D CT scan of a patient, it can take time to measure a 3D NMR spectrum, it can take time to conduct a 3D seismic survey.
In all cases, it would be highly desirable to obtain acceptable-quality reconstruction of the desired image, spectrum, or slice with fewer measurements and consequently less measurement time. The benefits could translate into less cost, more system throughput, and perhaps less radiation dosage, depending on the specifics of the system under study and the sensor characteristics.
In each of such fields there is a currently-accepted practice for the number of measurements required to obtain a faithful rendering of the object of interest (see F. Natterer, The Mathematics of Computerized Tomography, J. Wiley (1986); J. R. Higgins, Sampling theory in Fourier and signal analysis: foundations, Oxford: N.Y.: Clarendon Press; Oxford University Press (1996)). This accepted number of measurements sets the scale for the effort and cost required to obtain adequate information.